Scientific

Cell migration plays a central roles in embryonic development, wound healing and immune surveillance. In 2008, Yoichiro Mori, Alexandra Jilkine and LEK published a reaction-diffusion model for the initial step of cell migration, the front-back chemical polarization that sets a cell's directionality. (More detailed mathematical properties of this model were described by the same group in 2011.) Since then, progress has been made in investigating how that simple "wave-pinning" mechanism is shaped and tuned by feedback from other proteins, from the cell's environment (extracellular matrix), from interplay with larger signaling networks, and from cell-cell interactions. In this talk, we will describe some of this progress and mathematical questions that arise. In particular, AB will demonstrate how his numerical PDE bifurcation analysis has helped us to understand how cells repolarize to reverse their direction of motion.

Any countable group G gives rise to a von Neumann algebra L(G). The classification of these group von Neumann algebras is a central theme in operator algebras. I will survey recent rigidity results which provide instances when various algebraic properties of groups, such as the presence or absence of a direct product decomposition, are remembered by their von Neumann algebras. I will also explain the strongest such rigidity results, where L(G) completely remembers G, and discuss some of the open problems in the area.

Mirzakhani gave an inductive procedure to build random hyperbolic surfaces by gluing together smaller random pieces along curves. She proved that as the length of the gluing curve grows, these families equidistribute in the moduli space of hyperbolic surfaces. In this talk, I’ll explain how the conjugacy (exposited in James’s talk) between the earthquake and horocycle flows provides a template for translating equidistribution results for flat surfaces into equidistribution results for hyperbolic ones. Using this correspondence, we address Mirzakhani’s twist torus conjecture and exhibit new limiting distributions for hyperbolic surfaces built out of symmetric pieces. This is joint work (in progress) with James Farre.

A measured geodesic lamination on a hyperbolic surface encodes the
horizontal trajectory structure of certain quadratic differentials.
Thurston’s earthquake flow along such a lamination induces a dynamical
system on the moduli space of hyperbolic surfaces sharing many properties
with the classical Teichmüller horocycle flow. Mirzakhani gave a dynamical
correspondence between the earthquake and horocycle flows, defined
Lebesgue-almost everywhere. In this talk, we extend Mirzakhani’s conjugacy
and define an extension of the earthquake flow to an action of the upper
triangular group P in PSL(2,R) mapping certain flow lines to Teichmüller
geodesics. We classify the P-invariant ergodic probability measures as
those coming from affine invariant measures on quadratic differentials and
show that our map is a measurable isomorphism between P actions with
respect to these measures. This is joint work with Aaron Calderon.

We consider a stochastic bistable two-species generalized Lotka-Volterra model of the microbiome and use it as a testbed to analytically and numerically explore the role of direct (e.g., fecal microbiota transplantation) and indirect (e.g., changes in diet) bacteriotherapies. Two types of noise are included in this model, representing the immigration of bacteria into and within the gut (additive noise) and variations in growth rate associated with the spatially inhomogeneous distribution of resources (multiplicative noise). The efficacy of a bacteriotherapy is determined by comparing the mean first-passage times (the average time required for the system to transition from one basin of attraction to the other) with and without the intervention. Concepts from transition path theory are used to investigate how the role of noise affects these bacteriotherapies.